The Stacks project

Lemma 20.7.2. Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $U \subset X$ be an open subspace. Let $n > 0$ and let $\xi \in H^ n(U, \mathcal{F})$. Then there exists an open covering $U = \bigcup _{i\in I} U_ i$ such that $\xi |_{U_ i} = 0$ for all $i \in I$.

Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then

\[ H^ n(U, \mathcal{F}) = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ n(U) \to \mathcal{I}^{n + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^ n(U))}. \]

Pick an element $\tilde\xi \in \mathcal{I}^ n(U)$ representing the cohomology class in the presentation above. Since $\mathcal{I}^\bullet $ is an injective resolution of $\mathcal{F}$ and $n > 0$ we see that the complex $\mathcal{I}^\bullet $ is exact in degree $n$. Hence $\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1} \to \mathcal{I}^ n) = \mathop{\mathrm{Ker}}(\mathcal{I}^ n \to \mathcal{I}^{n + 1})$ as sheaves. Since $\tilde\xi $ is a section of the kernel sheaf over $U$ we conclude there exists an open covering $U = \bigcup _{i \in I} U_ i$ such that $\tilde\xi |_{U_ i}$ is the image under $d$ of a section $\xi _ i \in \mathcal{I}^{n - 1}(U_ i)$. By our definition of the restriction $\xi |_{U_ i}$ as corresponding to the class of $\tilde\xi |_{U_ i}$ we conclude. $\square$

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